Two vectors `5hat(i) + 7hat(j) - 3hat(k) and 2 hat(i) + 2hat(j) - a hat(k)` are mutually perpendicular . What is the value of a ?

Explanation

To find the value of \( a \) such that the vectors \( \mathbf{A} = 5\hat{i} + 7\hat{j} - 3\hat{k} \) and \( \mathbf{B} = 2\hat{i} + 2\hat{j} - a\hat{k} \) are mutually perpendicular, we can follow these steps:

### Step 1: Understand the Condition for Perpendicular Vectors
Two vectors are perpendicular if their dot product is zero. Therefore, we need to find \( \mathbf{A} \cdot \mathbf{B} = 0 \).

### Step 2: Write the Dot Product
The dot product of the vectors \( \mathbf{A} \) and \( \mathbf{B} \) can be calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (5\hat{i} + 7\hat{j} - 3\hat{k}) \cdot (2\hat{i} + 2\hat{j} - a\hat{k}) \]

### Step 3: Calculate the Dot Product
Using the properties of the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (5 \cdot 2) + (7 \cdot 2) + (-3 \cdot -a) \]
Calculating each term:
- \( 5 \cdot 2 = 10 \)
- \( 7 \cdot 2 = 14 \)
- \( -3 \cdot -a = 3a \)

Combining these results:
\[ \mathbf{A} \cdot \mathbf{B} = 10 + 14 + 3a = 24 + 3a \]

### Step 4: Set the Dot Product to Zero
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ 24 + 3a = 0 \]

### Step 5: Solve for \( a \)
Rearranging the equation:
\[ 3a = -24 \]
Dividing both sides by 3:
\[ a = -8 \]

### Final Answer
The value of \( a \) is \( -8 \).

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